The field of non-imaging optics has long sought a method to design optical surfaces that transform an incident light emitted by a light source into an arbitrary illumination pattern. In the last two decades, substantial progress has been made for the zero-étendue case, an idealization where light rays are exactly parallel or exactly diverging from a single point. This idealization allows a one-to-one correspondence between rays in the emitted light and rays in the target illumination pattern. This one-to-one correspondence reduces the design problem to determining an optical surface whose reflections or refractions implement a one-to-one mapping between the spatial density of rays in a cross-section of the emitted light and the spatial density of rays in the target illumination pattern. If a smooth mapping is possible between the initial and target densities, which is almost always the case for the zero-etendue systems, then that mapping can be found using the methods borrowed from the field of optimal mass transport. The resulting optics can produce detailed illumination patterns, for example, projecting photographic images. These optical surfaces are generally denoted as freeform optical surfaces, simply because their shapes are more complicated that any of the simple algebraic surfaces typically associated with lenses and mirrors.
In reality, the zero-étendue light source do not exist. Practical light sources, e.g., light-emitting diodes (LED), have spatial extent, i.e., light rays are emitted from an area, not a point, and these rays cross during their propagation, making one-to-one mappings impossible, and pushing the problem outside the scope of what optimal mass transport can solve. If a freeform optical surface is illuminated by a spatially extended light source, the resulting illumination pattern is significantly blurred, much as a shadow on a cloudy day becomes soft and indistinct. According to the second law of thermodynamics, this blurring is inescapable.
Consequently, when freeform optics are designed for spatially extended light sources, optical engineers have much more modest goals typically just achieving approximately uniform illumination in a circular or polygonally bounded region. Furthermore, it is usually accepted that there will be a blurry halo of uncontrolled illumination fall-off outside this region, even though that can be undesirable in some applications. Some researchers also seek to control this halo and achieve a sharp fall-off. The method of Simultaneous Multiple Surfaces (SMS) offers some control of the boundary by routing rays from the edge of the light source to predetermined targets; optimal mass transport combined with approximate deblurring can sometimes achieve sharp edges in the irradiation pattern. However, in both approaches, the final irradiation pattern suffers from an uncontrolled trade-off between blurry edges and undesirable texture artifacts inside the irradiation pattern.
The problem of obtaining a uniform irradiance from an Lambertian source has received much attention since the advent of high-powered LEDs. To date, all design methods are approximate. Furthermore, many of the methods proposed to design freeform surfaces rely on simplifying assumptions about the light source, most commonly, that it provides uniform flux through the lens. Most modern light sources are Lambertian, with flux intensity along any ray proportional to the cosine of the angle between that ray and the optical axis. This has to be modeled very carefully in the optimization, else the irradiance image has quite noticeable artifacts.
Accordingly, there is a need for methods that can transform incident light from the spatially extended light source into a target illumination pattern with sharp edges. Those methods can be beneficial for a number of optical applications, such as optics for signage illumination and specialized task lighting.